What's the best way to troubleshoot the stability of bitcoind?
I would say logging.
You could also use Linux tools to monitor usage of memory and other resources. Maybe vmstat or top would show something interesting. Maybe bitcoind is filling a filesystem you didn't expect - df -Pk.
The debug.log doesn't show anything particularly interesting.
If you are not already running bitcoind with option -debug=1, I suggest you try it.
-debug=<category>
Output debugging information (default: -nodebug, supplying <category> is
optional). If <category> is not supplied or if <category> = 1, output all
debugging information. <category> can be: net, tor, mempool, http, bench,
zmq, walletdb, rpc, estimatefee, addrman, selectcoins, reindex, cmpctblock,
rand, prune, proxy, mempoolrej, libevent, coindb, qt, leveldb, validation.
Note that the default is -nodebug.
I've had it just stop for no discernable reason.
As well as turning up debug to the max, I would try to capture the return code ($?) of bitcoind.
If i want to make sure my node is up... what's the best way to do that?
The daemon process startup system on your Linux distro ought to be able to auto-restart a stopped daemon. You configure the startup system appropriately. Details probably depend in distro and version. E.g. in systemd: restart=always